Finite Group Databasebeta

← #373351

Subgroups — #373351

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21178C211yesmaximal, characteristic, complemented
2#21098C211yesmaximal, characteristic
3C2×#1938C211yesmaximal, characteristic
4#20852C211yesmaximal, characteristic
5#20422C211yesmaximal, characteristic
6#11275C211yesmaximal, characteristic, complemented
7#11240C211yesmaximal, characteristic, complemented
8#11114C211yesmaximal, characteristic
9#11108C211yesmaximal, characteristic
10#10901C211yesmaximal, characteristic, complemented
11#10898C211yesmaximal, characteristic, complemented
12#10898C211yesmaximal, characteristic, complemented
13#10865C211yesmaximal, characteristic, complemented
14#10780C211yesmaximal, characteristic
15#10707C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21#2905C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#421C2211yescharacteristic
24C2×#421C2211yescharacteristic
25C22×#104C2211yescharacteristic
26C2×#416C2211yescharacteristic
27#2879C2211yescharacteristic
28C2×#408C2211yescharacteristic
29#2873C2211yescharacteristic
30C2×#405C2211yescharacteristic
31C2×#405C2211yescharacteristic
32C22×#97C2211yescharacteristic
33#1950—24no—
34#1938—22no—
35#1938C418yescomplemented
36#1937—22no—
37#1855—24no—
38#1765—22no—
39#1764—22no—
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42C2×#344C2211yescharacteristic
43C2×#344C2211yescharacteristic
44#1738C2211yescharacteristic
45#1738C2211yescharacteristic
46C2×#341C2211yescharacteristic
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51#1733C2211yescharacteristic
52#1733C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
58C22×#41C2311yescharacteristic
59#568—22no—
60#568—22no—
61#568—22no—
62#568—41no—
63C2×#131—22no—
64C2×#131—41no—
65C22×M16—22no—
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#430—22no—
81#430—22no—
82#430—22no—
83#430—41no—
84C2×#108—22no—
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—41no—
91C2×#108—41no—
92#424—22no—
93#424—22no—
94#424—41no—
95#424C2×C414yes—
96#423—22no—
97C2×C4⋊C8—22no—
98#421—22no—
99#421—22no—
100#421—22no—
101#421—41no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—41no—
105C2×#104—41no—
106C2×#104C2×C414yes—
107#416C2×C414yes—
108#416D412yes—
109#416D412yes—
110C2×#103Q812yes—
111C2×#102Q812yes—
112#408—22no—
113#408—22no—
114#408—41no—
115#408C2×C414yes—
116#407—22no—
117C2×C22⋊C8—22no—
118#405—22no—
119#405—22no—
120#405—22no—
121#405—41no—
122#405—41no—
123#405—41no—
124C4×M16—22no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—41no—
128C2×#97—41no—
129C2×#97C2×C414yes—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132#344—22no—
133#344—22no—
134#344—22no—
135#344—41no—
136#341—22no—
137#341—22no—
138#341—21no—
139#341—21no—
140#341—21no—
141#341—21no—
142#341C2×C414yes—
143#341D414yes—
144#340—22no—
145#339—22no—
146#339—22no—
147#339—22no—
148#339—22no—
149#339C2×C414yes—
150#336—41no—
151#336—41no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41C22×C412yes—
163C2×#41C2×D412yes—
164C22×Q8C2×Q811yescharacteristic
165C22×D4—21no—
166C22×D4C2×Q811yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C2411yescharacteristic
170#131—28no—
171#131—44no—
172#131—44no—
173#131—44no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—42no—
215C2×M16—44no—
216C2×M16—44no—
217C2×M16C22×C412yes—
218C2×M16C22×C412yes—
219C2×M16C2×D412yes—
220C2×M16C2×D412yes—
221C2×M16#4112yes—
222C2×M16#4112yes—
223C2×M16#4112yes—
224C2×M16#4112yes—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—42no—
242C22×C8C22×C412yes—
243C22×C8C22×C412yes—
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D412yes—
247C22×C8C2×Q812yes—
248C22×C8#4112yes—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117C22×C412yes—
258#117C2×D412yes—
259#117#4112yes—
260#117#4112yes—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4C2×Q812yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42C22×C412yes—
276#108—44no—
277#108—44no—
278#108—44no—
279#108—44no—
280C4⋊C8—28no—
281C4⋊C8—44no—
282#104—28no—
283#104—44no—
284#104—44no—
285#103C4⋊C414yes—
286#103C4⋊C414yes—
287#102C4⋊C414yes—
288#102C4⋊C414yes—
289C22⋊C8—28no—
290C22⋊C8—44no—
291#97—44no—
292#97—42no—
293#97—42no—
294#97—42no—
295C4×C8—42no—
296C4×C8—42no—
297C4×C8—42no—
298C24—21no—
299C242−1+411yescharacteristic
300#41—22no—
301#41—22no—
302#41—22no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8C2×C4⋊C412yes—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4C2×C4⋊C412yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—82no—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×Q811yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—22no—
371M16—22no—
372M16—22no—
373M16—22no—
374M16—22no—
375M16—48no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16#12414yes—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—48no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8C2×C4⋊C412yes—
435C2×C8C2×C4⋊C412yes—
436C2×C8C4×D414yes—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×C4⋊C414yes—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C23—22no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—82no—
464C23#52011yescharacteristic
465C23#53911yescharacteristic
466C23#54211yescharacteristic
467C23#54711yescharacteristic
468C23D4×Q811yescharacteristic
469C23#55211yescharacteristic
470C23#55611yescharacteristic
471Q8—22no—
472Q8—22no—
473Q8—41no—
474Q8—41no—
475D4—22no—
476D4—22no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—41no—
482D4—41no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4C22×C4⋊C411yescharacteristic
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C4×#4111yescharacteristic
503C2×C4C4×#4111yescharacteristic
504C2×C4C4×#4111yescharacteristic
505C2×C4#51811yescharacteristic
506C2×C4#52511yescharacteristic
507C2×C4#52511yescharacteristic
508C2×C4C2×#12411yescharacteristic
509C2×C4#52911yescharacteristic
510C2×C4#52911yescharacteristic
511C2×C4#53211yescharacteristic
512C2×C4#53311yescharacteristic
513C2×C4#53311yescharacteristic
514C2×C4#53811yescharacteristic
515C2×C4#54011yescharacteristic
516C2×C4D4211yescharacteristic
517C2×C4#54611yescharacteristic
518C2×C4#54611yescharacteristic
519C2×C4#54611yescharacteristic
520C2×C4#54611yescharacteristic
521C2×C4#55111yescharacteristic
522C2×C4#55311yescharacteristic
523C2×C4#55311yescharacteristic
524C2×C4#55311yescharacteristic
525C2×C4#55411yescharacteristic
526C2×C4#55411yescharacteristic
527C2×C4#55411yescharacteristic
528C8—22no—
529C8—22no—
530C8—24no—
531C8—24no—
532C8—42no—
533C8—42no—
534C8—42no—
535C8—42no—
536C8—42no—
537C8—81no—
538C8—81no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—44no—
548C22—161no—
549C22—161no—
550C22#233511yescharacteristic
551C22D4×C4⋊C411yescharacteristic
552C22#237311yescharacteristic
553C22#246611yescharacteristic
554C22#247011yescharacteristic
555C22#249211yescharacteristic
556C22#255611yescharacteristic
557C4—42no—
558C4—44no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4#229611yescharacteristic
563C4C4×#12411yescharacteristic
564C4#234911yescharacteristic
565C4#236411yescharacteristic
566C4#246411yescharacteristic
567C4#247211yescharacteristic
568C4#252211yescharacteristic
569C4#258711yescharacteristic
570C2—21no—
571C2—21no—
572C2—21no—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—82no—
577C2#1411611yescharacteristic
578C2#2269512yes—
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22D4×C4⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21178C2semidirect
6. #11275C2semidirect
7. #11240C2semidirect
10. #10901C2semidirect
11. #10898C2semidirect
12. #10898C2semidirect
13. #10865C2semidirect
15. #10707C2semidirect
35. #1938C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#211782
#210982
C2×#19382
#208522
#204222
#112752
#112402
#111142
#111082
#109012
#108982
#108652
#107802
#107072